This river has 37 bridges just in Paris, so you can enjoy the entire mystique across the city.
A great way to take advantage of the river is to book a mini cruise, I recommend the one at night that includes dinner and 2 hours on the ship. It a great way to see the city at night, with all the lights. The prices start at 50$.
One of the most famous bridges is Pont de l'Archevêché, lovers place their padlocks as a symbol of their union. Unfortunately the grids with the padlock will be removed soon because the bridge doesn’t support all that weight.